Exponents and powers: complete GMAT guide

For positive bases $a$ and $b$, and real exponents $m$ and $n$:

These rules apply when the bases are the same or when the exponent is distributed over a product or quotient. They do not apply to sums: $(a+b)^2 \neq a^2+b^2$.

There is no simple rule for adding powers with the same base. Factor the common power instead. For example,

and

This factoring move appears often in questions that ask for the value of an expression or the largest power of a number that divides a sum.

A negative exponent means a reciprocal, not a negative result. $3^{-2}=1/9$, which is positive. Be careful with expressions such as $(-3)^{-2}$: the base is negative, but the square makes it positive, so the result is $1/9$. By contrast, $-3^{-2}=-(1/9)$.

Any nonzero number raised to the zero power is 1. The expression $0^0$ is undefined on the GMAT, and a base of zero with a negative exponent is also undefined.

A fractional exponent combines a power and a root. In general,

For example, $8^{2/3}=(\sqrt[3]{8})^2=2^2=4$. Take the root first when the root is easy, because the numbers stay smaller.

A negative fractional exponent means the reciprocal of the positive fractional power: $a^{-m/n}=1/a^{m/n}$.

To compare powers, try to write them with the same base or the same exponent. For example, compare $2^{30}$ and $3^{20}$. Rewrite both with exponent 10: $2^{30}=(2^3)^{10}=8^{10}$ and $3^{20}=(3^2)^{10}=9^{10}$. Since $9 \gt 8$, $3^{20}$ is larger.

When the bases are the same, the larger exponent gives the larger value if the base is greater than 1. If the base is between 0 and 1, the order reverses: $(1/2)^3 \lt (1/2)^2$.

Questions about the units digit of a large power use repeating cycles. The units digit of powers of 2 repeats 2, 4, 8, 6. For powers of 3, the cycle is 3, 9, 7, 1. For powers of 7, it is 7, 9, 3, 1. Divide the exponent by the cycle length and use the remainder to locate the digit in the cycle.

To solve an equation such as $2^{x+1}=32$, write both sides with the same base: $2^{x+1}=2^5$, so $x+1=5$ and $x=4$. If the bases cannot be made equal, look for a substitution or a common factor.

For equations with addition, factor out the smallest power. For example, $2^x+2^{x+1}=48$ becomes $2^x(1+2)=48$, so $2^x=16$ and $x=4$.

Exponent questions often appear inside word problems about growth, decay, or repeated doubling. If a quantity doubles every hour, its value after $t$ hours is $P \cdot 2^t$, where $P$ is the starting value. If it halves every hour, the multiplier is $(1/2)^t$. The exponent counts the number of intervals, not the total time. If a quantity doubles every 30 minutes and the total time is 3 hours, the exponent is 6, not 3.

Decay problems use a multiplier between 0 and 1. If a substance loses half of its mass each year, the remaining mass after $t$ years is $P(1/2)^t$. If it loses 20% each year, the multiplier is 0.8, and the remaining mass is $P(0.8)^t$. Distinguish between "loses half" and "loses 50% of the current amount." The first is a multiplier of 1/2. The second is the same in this case, but the wording can change the base in more complex problems.

When an exponent rule feels uncertain, test it with small numbers. To check whether $(a^m)^n=a^{mn}$, try $(2^2)^3=4^3=64$ and $2^6=64$. To check whether $a^m a^n=a^{m+n}$, try $2^2 \cdot 2^3=4 \cdot 8=32$ and $2^5=32$. Small cases do not prove a rule, but they catch misremembered ones.

Watch for the difference between a power of a product and a product of powers. $(2x)^3=8x^3$, but $2x^3$ means $2 \cdot x^3$. The parentheses decide whether the coefficient is raised to the power. This distinction appears often in algebra questions that mix exponents with variables.

Finally, check the sign of the base. A negative base raised to an even power is positive, and a negative base raised to an odd power is negative. $(-2)^4=16$, while $(-2)^5=-32$. If the negative sign is outside the parentheses, it is not part of the base: $-2^4=-16$. Write the base in parentheses before you apply the exponent, and the sign will take care of itself.

1. Write every term with the smallest possible base.
2. Use the laws of exponents to combine terms with the same base.
3. For sums or differences, factor out the common power.
4. For comparisons, match the base or the exponent.
5. For units digits, find the cycle and use the remainder.
6. Check whether the base is zero or negative, since the rules change.

• Adding exponents when bases are multiplied but exponents are not the same.
• Distributing an exponent over a sum: $(x+y)^2 \neq x^2+y^2$.
• Treating a negative exponent as a negative number.
• Forgetting that $a^0=1$ for nonzero $a$.
• Applying the product rule to terms with different bases.
• Mixing up $(-2)^4$ and $-2^4$.
• Using the wrong position in a repeating units-digit cycle.